GRAVITON PRODUCTION IN ELLIPTIC AND HYPERBOLIC UNIVERSES

The problem of cosmological graviton creation for homogeneous and isot ropic universes with elliptical (epsilon=+1) and hyperbolical (epsilon =-1) geometries is addressed. The gravitational wave equation is estab lished for a self-gravitating fluid satisfying the barotropic equation of state p=(gamma-1)rho, which is the source of Einstein's equations plus a cosmological Lambda term. The time-dependent part of this equat ion is exactly solved in terms of hypergeometric functions for any val ue of gamma and spatial curvature epsilon. An expression representing ari adiabatic vacuum state is then obtained in terms of associated Leg endre functions whenever gamma not equal 2/3 [(2n+1)/(2n-1)], where n is an integer. This includes most cases of physical interest such as g amma=0, 4/3,1. The mechanism of graviton creation is reviewed and the Bogoliubov coefficients related to transitions between arbitrary cosmi c eras are also explicitly evaluated.